Chapter 9

Assume that \(S\) forms the boundary of a closed and bounded region \(D\). If \(\mathbf{F}=P \mathbf{i}+Q \mathbf{j}+R \mathbf{k}\) and \(P, Q\), and \(R\) have continuous second partial derivatives, prove that $$ \iint_{S}(\operatorname{curl} \mathbf{F} \cdot \mathbf{n}) d S=0 . $$

Consider the surface integral \(\iint_{S}(\) curl \(\mathbf{F}) \cdot \mathbf{n} d S\), where \(\mathbf{F}=x y z \mathbf{k}\) and \(S\) is that portion of the paraboloid \(z=1-x^{2}-y^{2}\) for \(z \geq 0\) oriented upward. (a) Evaluate the surface integral by the method of Section 9.13; that is, do not use Stokes' theorem. (b) Evaluate the surface integral by finding a simpler surface that is oriented upward and has the same boundary as the paraboloid. (c) Use Stokes' theorem to verify the result in part (b).

Find the indicated moment of inertia of the lamina that has the given shape and density. $$ r=a ; \rho(r, \theta)=\frac{1}{1+r^{4}} ; I_{x} $$

Let \(\mathbf{a}\) be a constant voctor and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. \(\operatorname{curl} \mathbf{r}=\mathbf{0}\)

Find the directional derivative of the given function at the given point in the indicated direction. $$ F(x, y, z)=\frac{x^{2}-y^{2}}{z^{2}} ;(2,4,-1), \mathbf{i}-2 \mathbf{j}+\mathbf{k} $$

(a) Find the curvature of an elliptical orbit that is described by \(\mathbf{r}(t)=a \cos t \mathbf{i}+b \sin t \mathbf{j}+c \mathbf{k}, a>0, b>0, c>0\). (b) Show that when \(a=b\), the curvature of a circular orbit is the constant \(\kappa=1 / a\).

Find \(\mathbf{r}^{\prime}(t)\) and \(\mathbf{r}^{\prime \prime}(t)\) for the given vector function. $$ \mathbf{r}(t)=\langle t \cos t-\sin t, t+\cos t\rangle $$

Sketch the region \(D\) whose volume \(V\) is given by the iterated integral. $$ \int_{0}^{2} \int_{0}^{3-y} \int_{-\sqrt{y}}^{\sqrt{y}} d x d z d y $$

a) Find the curvature of an elliptical orbit that is describeo by \(\mathbf{r}(t)=a \cos t \mathbf{i}+b \sin t \mathbf{j}+c \mathbf{k}, a>0, b>0, c>0\) b) Show that when \(a=b\), the curvature of a circular orbit is the constant \(\kappa=1 / a\)

Evaluate the surface integral \(\iint_{S} G(x, y, z) d S\). \(G(x, y, z)=\left(x^{2}+y^{2}\right) z ; S\) that portion of the sphere \(x^{2}+y^{2}+z^{2}=36\) in the first octant

Evaluate the double integral over the region \(R\) that is bounded by the graphs of the given equations. Choose the most convenient order of integration. $$ \iint_{R} \frac{y}{1+x y} d A ; y=0, y=1, x=0, x=1 $$

Let a be a constant vector and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ (\mathbf{a} \times \nabla) \times \mathbf{r}=-2 \mathbf{a} $$

Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ x z=6 ;(2,0,3) $$

Find the first partial derivatives of the given function. $$ z=\left(x^{3}-y^{2}\right)^{-1} $$

In Problems, find \(\mathbf{r}^{\prime}(t)\) and \(\mathbf{r}^{\prime \prime}(t)\) for the given vector function. \(\mathbf{r}(t)=\left\langle t e^{2 t}, t^{3}, 4 t^{2}-t\right\rangle\)

In Problems, find the directional derivative of the given function at the given point in the indicated direction. $$ \begin{aligned} &F(x, y, z)=\sqrt{x^{2} y+2 y^{2} z} ;(-2,2,1), \text { in the direction of the }\\\ &\text { negative } z \text { -axis } \end{aligned} $$

Evaluate the given integral by means of the indicated change of variables. \(\iint_{R} \frac{x}{y+x^{2}} d A\), where \(R\) is the region in the first quadrant bounded by the graphs of \(x=1, y=x^{2}, y=4-x^{2}\); \(x=\sqrt{v-u}, y=v+u\)

Assume that \(f\) and \(g\) are scalar functions with continuous second partial derivatives. Use the divergence theorem to establish Green's identities. $$ \iint_{S}(f \nabla g) \cdot \mathbf{n} d S=\iiint_{D}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V $$

Show that the given integral is independent of the path. Evaluate. $$ \int_{(1,1,1)}^{(2,4,8)} y z d x+x z d y+x y d z $$

Let \(\mathbf{a}\) be a constant voctor and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \nabla \times(\mathbf{a} \times \mathbf{r})=2 \mathbf{a} $$

Show that the curvature of a straight line is the constant \(\kappa=0\).

Find \(\mathbf{r}^{\prime}(t)\) and \(\mathbf{r}^{\prime \prime}(t)\) for the given vector function. $$ \mathbf{r}(t)=\left\langle t e^{2 t}, t^{3}, 4 t^{2}-t\right\rangle $$

Evaluate the given integral by means of the indicated change of variables. \(\iint_{R} y d A\), where \(R\) is the triangular region with vertices \((0,0)\), \((2,3)\) and \((-4,1) ; x=2 u-4 v, y=3 u+v\)

In Problems, find the indicated moment of inertia of the lamina that has the given shape and density. $$ \begin{aligned} &\text { Outside } r=1 \text { and inside } r=2 \sin 2 \theta, \text { first quadrant; }\\\ &\rho(r, \theta)=\sec ^{2} \theta ; I_{y} \end{aligned} $$

Sketch the region \(D\) whose volume \(V\) is given by the iterated integral. $$ \int_{1}^{3} \int_{0}^{1 / x} \int_{0}^{3} d y d z d x $$

Find the curvature at \(t=\pi\) of the cycloid that is described by $$ \mathbf{r}(t)=a(t-\sin t) \mathbf{i}+a(1-\cos t) \mathbf{j}, a>0 $$

Evaluate the surface integral \(\iint_{S} G(x, y, z) d S\). \begin{aligned} &G(x, y, z)=z^{2} ; S \text { that portion of the plane } z=x+1 \text { within }\\\ &\text { the cylinder } y=1-x^{2}, 0 \leq y \leq 1 \end{aligned}

Evaluate the double integral over the region \(R\) that is bounded by the graphs of the given equations. Choose the most convenient order of integration. $$ \iint_{R} \sin \frac{\pi x}{y} d A ; x=y^{2}, x=0, y=1, y=2 $$

In Problems, show that the given integral is independent of the path. Evaluate. $$ \int_{(0,0,0)}^{(1,1,1)} 2 x d x+3 y^{2} d y+4 z^{3} d z $$

Let a be a constant vector and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \nabla \times(\mathbf{a} \times \mathbf{r})=2 \mathbf{a} $$

Find the first partial derivatives of the given function. $$ z=\left(-x^{4}+7 y^{2}+3 y\right)^{6} $$

In Problems, find \(\mathbf{r}^{\prime}(t)\) and \(\mathbf{r}^{\prime \prime}(t)\) for the given vector function. \(\mathbf{r}(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j}+\tan ^{-1} t \mathbf{k}\)

In Problems, find the directional derivative of the given function at the given point in the indicated direction. \(F(x, y, z)=2 x-y^{2}+z^{2} ;(4,-4,2)\), in the direction of the origin

Fill in the blank or answer true/false. Where appropriate, assume continuity of \(P, O\), and their first partial derivatives. If \(\mathbf{F}=f(x) \mathbf{i}+g(y) \mathbf{j}+h(z) \mathbf{k}\), then curl \(\mathbf{F}=\)____

Find the indicated moment of inertia of the lamina that has the given shape and density. Outside \(r=1\) and inside \(r=2 \sin 2 \theta\), first quadrant; \(\rho(r, \theta)=\sec ^{2} \theta ; I_{y}\)

Show that the given integral is independent of the path. Evaluate. $$ \int_{(0,0,0)}^{(1,1,1)} 2 x d x+3 y^{2} d y+4 z^{3} d z $$

Let \(\mathbf{a}\) be a constant voctor and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \nabla \times(\mathbf{a} \times \mathbf{r})=2 \mathbf{a} $$

Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ x z=6 ;(2,0,3) $$

Find \(\mathbf{r}^{\prime}(t)\) and \(\mathbf{r}^{\prime \prime}(t)\) for the given vector function. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j}+\tan ^{-1} t \mathbf{k} $$

In Problems \(21-24\), find the volume of the solid bounded by the graphs of the given equations. \(x=y^{2}, 4-x=y^{2}, z=0, z=3\)

Let \(C\) be a plane curve traced by \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) j\), where \(f\) and \(g\) have secood derivatives. Show that the curvature at a point is given by $$ \kappa=\begin{aligned} &\left|f^{\prime}(t) g^{\prime \prime}(t)-g^{\prime}(t) f^{\prime \prime}(t)\right| \\ &\left(\left[f^{\prime}(t)\right]^{2}+\left[g^{\prime}(t)\right]^{2}\right)^{3 / 2} \end{aligned} $$

Evaluate the surface integral \(\iint_{S} G(x, y, z) d S\). \(G(x, y, z)=x y ; S\) that portion ofthe paraboloid \(2 z=4-x^{2}-y^{2}\) within \(0 \leq x \leq 1,0 \leq y \leq 1\)

In Problems, show that the given integral is independent of the path. Evaluate. $$ \int_{(1,0,0)}^{(2, \pi / 2,1)}\left(2 x \sin y+e^{3 z}\right) d x+x^{2} \cos y d y+\left(3 x e^{3 z}+5\right) d z $$

Let a be a constant vector and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \nabla \cdot(\mathbf{a} \times \mathbf{r})=0 $$

Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ z=\cos (2 x+y) ;(\pi / 2, \pi / 4,-1 / \sqrt{2}) $$

Find the first partial derivatives of the given function. $$ z=\cos ^{2} 5 x+\sin ^{2} 5 y $$

In Problems, consider the plane through the points \(P\) and \(Q\) that is perpendicular to the \(x y\) -plane. Find the slope of the tangent at the indicated point to the curve of intersection of this plane and the graph of the given function in the direction of \(Q\). $$ f(x, y)=(x-y)^{2} ; P(4,2), Q(0,1) ;(4,2,4) $$

Evaluate the given integral by means of the indicated change of variables. \(\iint_{R} y^{4} d A\), where \(R\) is the region in the first quadrant bounded by the graphs of \(x y=1, x y=4, y=x, y=4 x ; u=x y, v=y / x\)

Find the volume of the solid bounded by the graphs of the given equations. $$ x=y^{2}, 4-x=y^{2}, \quad z=0, \quad z=3 $$

Evaluate the double integral over the region \(R\) that is bounded by the graphs of the given equations. Choose the most convenient order of integration. $$ \iint_{R} \sqrt{x^{2}+1} d A ; x=y, x=-y, x=\sqrt{3} $$